Harder–Narasimhan stacks for principal bundles in higher dimensions and arbitrary characteristics

Let [Formula: see text] be a split reductive group over a field [Formula: see text] of arbitrary characteristic, chosen suitably. Let [Formula: see text] be a smooth projective morphism of locally noetherian [Formula: see text]-schemes with geometrically connected fibers, together with a relatively very ample line bundle [Formula: see text]. We show that for each Harder–Narasimhan type [Formula: see text] for principal [Formula: see text]-bundles, all pairs consisting of a principal [Formula: see text]-bundle on a fiber of [Formula: see text] together with a given canonical reduction of HN-type [Formula: see text] form an algebraic stack [Formula: see text] over [Formula: see text]. The forgetful [Formula: see text]-morphism [Formula: see text] to the algebraic stack of all principal [Formula: see text]-bundles on fibers of [Formula: see text] is a schematic morphism, which is of finite type, separated, radicial and induces an isomorphism on residue fields of all points of [Formula: see text]. It factors via an open substack [Formula: see text] of [Formula: see text], inducing a finite morphism [Formula: see text]. This is a closed embedding if the Behrend conjecture is satisfied by [Formula: see text]. Along the way we prove the openness of semistability and the semicontinuity of canonical type in a family. The results of this paper hold in all characteristics, and in fact it gives better proofs of the results of [S. Gurjar and N. Nitsure, Schematic Harder–Narasimhan stratification for families of principal bundles and [Formula: see text]-modules, Proc. Math. Sci. 124 (2014) 315–332] and [S. Gurjar and N. Nitsure, Schematic Harder–Narasimhan stratification for families of principal bundles in higher dimensions, Math. Z. 289 (2018) 1121–1142] which had assumed the characteristic to be zero.